A prime example of a type of business transaction that required facility with ratio and proportion is barter, the direct exchange of goods or services. Barter was still common in Europe in the late Middle Ages; one reason is that prior to the European encounter with the New World in 1492, precious metals (and therefore coins) were relatively scarce. The whole ninth chapter of Fibonacci’s Liber Abbaci was devoted to “barter of merchandise and similar things.”

At the same time, barter was beginning to coexist with systems of money that were becoming more common. A sign that barter had entered a new stage was that many barter transactions were being carried out on the basis of rendering the goods’ values in a common monetary unit: this many coins worth of wheat, that many coins worth of oats, etc. A barter was often recorded as such in a register or account book, as if actual coins had been exchanged, when in fact no coins at all were involved.

Because coins were still scarce, the widespread custom among merchants was to set the barter price for a commodity by “marking up” the cash price by a certain percentage. The two barterers had to agree on the markup rate ahead of time, or else one would feel cheated. This type of barter calculation is seen in the following problem drawn from the Pamiers manuscript.

Figure 7. Two merchants negotiate terms to barter wool for cloth in this drawing from an illuminated manuscript version of Filippo Calandri’s Trattato di Arithmetica, an abbaco treatise with many problems similar to those of the Pamiers manuscript. It was originallyprinted in Florence in 1491. (Source: Biblioteca Riccardiana (Florence, Italy), Ricc. 2669, page 66 recto. Image used by permission; further reproduction is prohibited.)

Problem 1. Two merchants want to barter together, planning to exchange their merchandise; and the first of them has wool and the other has cloth. The one who has cloth wants 24 sous for trading each bolt of cloth whose cash price is but 20 sous. I ask for how much the other must sell each quintal of wool that is worth but 12 liras, in order that he lose nothing. (Sesiano 1984, p. 57)

In order that the second trader “lose nothing,” he must augment his price of 12 liras by the same proportional amount that the first trader has augmented his price of 20 sous to become 24. Thus, the problem amounts to a simple question of ratio and proportion. The author of the manuscript showed how to solve it in two different ways, using the rule of three and double false position, respectively.

Notice that in the above problem, different trade goods were measured with different units: the cloth was measured in bolts, whereas raw wool (as well as other animal hides and many other types of commodities) was measured in quintals, i.e., hundredweights. Even worse, in Medieval times linen might be measured out with different units than other types of woven cloth; grains with different units than other types of food; etc. Expressing all commodity values in some common unit of currency, such as the sous or lira, helped bring some order amid the anarchy of physical units. In fact, this was one of the reasons for the adoption of systems of currency.

In the system of coins that had existed in France since the rule of Charlemagne in the 9th century, 20 sols or sous (from Latin solidus) were equal to one lira or livre (from Latin libra), literally a pound of silver. In English, the sol or sous became the shilling, and the lira or livre became the pound; this is why the British pound was always abbreviated £ for livre.

The Arte dell’Abbaco, an anonymous abbaco text published in the Venetian dialect in 1478, includes a virtually identical problem (see Swetz 1987, p. 152). Like the one above, the problem involves two merchants bartering wool for cloth, with the terms reckoned in soldi; only the numbers are different. Arte dell’Abbaco, the oldest known mathematics book printed in Europe, is often called the Treviso Arithmetic because it was printed in Treviso, just inland from Venice, Italy.